#132142

order	512
nilpotency class	4
isoclinism	isoclinic class of #132019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2312 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

491 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,32,42,5,26)(3,15,19,9,33,36,17,52)(4,21,24,55,57,39,28,41)(7,14,30,11,8,13,29,43)(10,22,38,25,56,59,60,23)(16,46,49,62,48,44,53,34)(18,40,31,35,20,45,27,37)(47,63,50,64,58,54,51,61);(1,3,8,20)(2,9,13,37)(4,16,23,51)(5,27,30,19)(6,31,29,17)(7,18,32,33)(10,34,21,54)(11,40,26,36)(12,15,43,45)(14,35,42,52)(22,53,24,58)(25,50,57,48)(28,47,59,49)(38,44,55,64)(39,63,56,62)(41,61,60,46);(1,4,8,23)(2,10)(3,16,20,51)(5,28,30,59)(6,24,29,22)(7,25,32,57)(9,34)(11,41)(12,38)(13,21)(14,39)(15,44)(17,53,31,58)(18,50,33,48)(19,49,27,47)(26,60)(35,63)(36,46)(37,54)(40,61)(42,56)(43,55)(45,64)(52,62)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39338);